Abstract
We extend the result of Kirk-Saliga and we generalize Alfuraidan and Khamsi theorem for reflexive graphs. As a consequence, we obtain the ordered version of Caristi’s fixed point theorem. Some concrete examples are given to support the obtained results.
1. Introduction
Fixed point theory is one of the most useful tools in mathematics; it is used to solve many existence problems such as differential equations, control theory, optimization, and several other branches (for the literature see [1]). The most well-known fixed point result is Banach contraction principle [2]; it is famous for its applications, proving the existence of solution of integral equations by converting the problem to fixed point problem (see [3]). Recall that a point is called a fixed point for a map if . Due to its importance, this theorem found a number of generalizations and extensions in many directions; for more details see [4] and the references therein. In 1976, Caristi (see [5]) gave an elegant generalization of Banach contraction principle, where the assumption that “ is continuous” is dropped and replaced by a weak assumption. Since then, various proofs, extensions, and generalizations are given by many authors (see [6–8]). It is worth mentioning that Caristi’s fixed point theorem is equivalent to the Ekeland variational principle [8]. Also, it characterizes the completeness of the metric space as showed by Kirk in [9]. Among those generalizations, there is Kirk-Saliga fixed point theorem (see [10]) which states that any map has a fixed point provided that is complete metric space and there exist an integer and a lower semicontinuous function such that and for any . For more on the latter result, one can consult [11].
Recently, Ran and Reurings [12] extend the Banach contraction principle in the context of partially ordered set where the contraction is restricted to the comparable elements which allowed them to give a meaningful application to linear and nonlinear matrix equations. Moreover, Nieto and Rodríguez-López in [13] have weakened the continuity assumption using a more suitable condition where the order is combined with the topological properties. For more details, one can consult [14, 15]. Also, in [16] Alfuraidan and Khamsi gave an analogue version of Caristi’s fixed point theorem in the setting of partially ordered metric space where the inequality holds only for comparable elements. However, the new approach in their work is mixing the concept of the reflexive acyclic digraph with fixed point results. In this article, we discuss an extension of Kirk-Saliga result and we generalize Alfuraidan and Khamsi theorem for reflexive graphs. As a corollary, we obtain the ordered version of Caristi’s fixed point theorem. Some concrete examples are given to support the obtained results. Throughout this paper we denote by the set of all integers and by the set of all positive integers.
2. Preliminaries
We start by recalling some basic notions on graphs borrowed from [17].
Definition 1. Let be an arbitrary set.(i)A directed graph, or digraph, is a pair , where is a subset of the Cartesian product . The elements of are called vertices or nodes of and the elements of are the edges also called oriented edges or arcs of . An edge of the form is a loop on . Another way to express that is a subset of is to say that is a binary relation over . Given a digraph , the set of vertices (of edges) of is denoted by ().(ii)The digraph is said to be transitive if whenever and , .
Definition 2. A digraph is said to be reflexive if is a subset of . Otherwise, every vertex has a loop.
Definition 3. Let be a digraph. (i)A vertex is said to be isolated if for all vertex , we have neither nor .(ii)Two vertices . A path in , from (or joining) to , is a sequence of vertices , such that , and , for all . The integer is the length of the path . If and , the path is called a directed cycle. An acyclic digraph is a digraph which has no directed cycle.(iii)We denote by the fact that can be reached from by means of a path in .
A metric space endowed with a digraph such that is denoted by . The following notion of regularity is borrowed from Alfuraidan and Khamsi in [16] that considered it for posets.
Definition 4. Let be a partially ordered metric space. We say that satisfies the condition (OSC) if for any decreasing sequence that is convergent to , .
In the setting of digraphs, the analogue of the infimum of chain may be stated as follows.
Definition 5. Let be metric space endowed with a digraph. We say that satisfies the condition (OSCL) if for any sequence that is convergent to and for all , , for all and if there exists such that , for all , then .
Remark 6. Let be a partially ordered metric space. Let be the digraph associated with the order (see [16]). One can see that Under the above observations, the (OSCL) property is reduced to the (OSC) condition.
Let be the first transfinite ordinal and let be the first uncountable transfinite ordinal. is the order type of “the set of integers” and is the order type of the set of real numbers. Note that, for each , is countable.
Proposition 7 (see [11]). The following is valid: (i)The ordinal cannot be attained via sequential limits of countable ordinals. That is if is an ascending sequence of countable ordinals, then the ordinal is countable too.(ii)Each second kind countable ordinal is attainable via such sequences. In other words: if is of second kind (ordinal limit), then there exists a strictly ascending sequence of countable ordinals with property (3).
The following result is needed throughout this work; for the proof see [18, Proposition A.6, pp. 284].
Proposition 8. Suppose that a sequence is bounded and either nonincreasing or nondecreasing. Then there exists such that for all .
We conclude this section by the following useful definitions.
Definition 9. Let be metric space endowed with a digraph, and a lower semicontinuous function. Let be a self-mapping. We say the following: (1) is a -monotone if for all , (2) is a -Caristi mapping if for all , (3) is a -Kirk-Saliga mapping if for all ,
3. Main Results
Theorem 10. Let be a complete metric space endowed with a reflexive digraph satisfying the (OSCL) condition. Let be a -monotone and -Kirk-Saliga mapping. If there exists an element such that , then admits a fixed point in .
Proof. If then , for all such that . Assume that and consider the function defined from into by The idea of the proof is to construct a transfinite orbit , where is the first uncountable ordinal satisfying, for each , : ; : , whenever is an ordinal limit; : , whenever ; : , whenever . Consider the sequence defined for each by . Since and using the monotony of , we obtain for each . According to (KS2), the nonnegative sequence is decreasing and then converges. From (KS1), we get that for all integers Hence, is a Cauchy sequence and then converges to . Let us put . Clearly the properties – are satisfied for each . Let . Assume that the orbit has been defined. We need to define and show that the four properties – hold. For that, we have to distinguish two cases, when is an immediate successor or is an ordinal limit. Clearly and are satisfied; let us focus on and .
Claim 1 (C holds)
Case 1. Assume that is an ordinal limit; that is, there exists a strictly ascending sequence of ordinals in such that and whenever . Since holds for all , we get which implies that is decreasing sequence in and hence it is convergent. Then is Cauchy sequence, so it converges in . Set . By (OSCL) property, we obtain for all . Let . There exists such that for each we have and thus for each , Since is taken arbitrary, we obtain .
Case 2. Assume that is an immediate successor; there exists such that . (i)If is an immediate successor, there exists an ordinal such . From , we have and using the -monotonicy of it follows that and so holds.(ii)If is an ordinal limit, from Proposition 7, there exists an ascending sequence such that . From we have . Using the (OSCL) condition, we have . Since is -monotone, and as , we get . Again, (OSCL) insures that . Then holds.Claim 2 (D holds)
Case 1. Assume that is ordinal limit. Let . There exists such that for each we have Then we get for each that and for all Since is lower semicontinuous, we get From , we have for all . Using the same argument as above, we get and (OSCL) insures that . Hence, for all , . This implies that By passing to limit superior in inequality (13), it follows that Hence, holds.
Case 2. Assume that is an immediate successor; we have shown above that holds. Then and by assumption we get and for all , we have The triangle inequality implies that for each , which completes the proof of in both cases.
Thus, the orbit is well constructed. Since is nonincreasing on and is uncountable, there must exist such that is constant for all . From , we get Hence, .
We support our result by giving an example of a mapping which is -Kirk-Saliga mapping, for some integer , but not -Caristi.
Example 11. Consider the metric space , where and , for all . Endow with the directed graph represented in Figure 1, where Consider the function defined by and the mapping defined by , if ; .
One can see that , and We verify the following assertions: (i) is complete and is -monotone obviously.(ii) satisfies the (OSCL) property. Indeed, let be a sequence in such that converges to some and , for all . Two cases to distinguish are as follows:(1)There exists such that , for all . Then for all , . If is an isolated vertex, the (OSCL) is obviously satisfied. If not, for all implies . Thus, (OSCL) is satisfied.(2)For all , there exists such . Then ; that is, there exists a nondecreasing function such that for all , and . If for all , then . Thus, (OSCL) is satisfied.(iii) is -kirk-Saliga mapping in with . Indeed, for all , but is not -Caristi mapping, since (iv) and admits a fixed point in which is 0.
If we remove the (OSCL) property, we are not certain that the fixed point will be obtained. Let us illustrate that by this counterexample.
Example 12. Replace in the above example the digraph by the digraph represented in Figure 2, where and we consider the mapping defined as follows: One can see that satisfies (OSC) property but does not satisfy the (OSCL), since and for all , and but . The mapping satisfies all others conditions of Theorem 10 but has no fixed point in .
Corollary 13. Let be a complete metric space endowed with a reflexive digraph which satisfies the (OSCL) property. Let , where is prime integer and , for each , and let be a -monotone mapping such that there exists , for all , Then admits a fixed point in provided that there exists an element such that .
Proof. Clearly, satisfies all conditions of Theorem 10; then there exists such that and so, . Since , we get from (29) thatNow, let ; then . By Bezout identity, there exists such that and . Without loss of generality, we suppose that and . Since and , then and . Since , then .
We conclude this work by a discussion about preordered sets.
Let be a preordered set; that is, the binary relation “” is reflexive and transitive.
Given a reflexive digraph , we can always define a preorder on as follows: Conversely, if is a preordered set, we define the reflexive digraph as follows: two vertices are connected by an arc from to if . Note that is transitive (i.e., if and ; then ), so . These remarks lead to the following definition.
Definition 14. Let be a preordered set. We say that satisfies the (OSCL) condition if and only if satisfies the (OSCL) condition.
We shall say that is -monotone (resp., -Kirk-Saliga) mapping if -monotone (resp., -Kirk-Saliga) mapping.
An analogue version of Theorem 10 in the setting of the preordered metric spaces may be stated as follows.
Theorem 15. Let be a preordered complete metric space satisfying the (OSCL) condition. Let be a -monotone and -Kirk-Saliga mapping. If there exists an element such that , then admits a fixed point in .
Remark 16. If moreover the above binary relation is antisymmetric (i.e., ( and ) imply ), we obtain, from Remark 6, the following result established by Alfuraidan and Khamsi.
Corollary 17 (see [16, Theorem 5]). Let () be a complete partially ordered metric space satisfying the property (OSC). Let be a -monotone and -Kirk-Saliga mapping with . If there exists an element such that , then admits a fixed point in .
Conflicts of Interest
The authors declare that they have no conflicts of interest.